Step by Step

Differential Equations · Axiom Academy

Follow the tangent line a short distance, land on a new point, then repeat — one straight step at a time. Watch, then drive it yourself. Solving a differential equation, one straight step at a time Most differential equations can't be solved by algebra alone. Euler's method sidesteps that: start at a known point, use the equation itself to find the slope there, walk a short straight distance along that slope, and repeat. No calculus tricks — just arithmetic, over and over. Watch Euler's method solve with step size h = 0.5 . Each red segment is a tangent-line step; the orange dots are where it lands. Notice how the path pulls away from the true solution (blue) as the steps add up. Each step follows a straight line where the true solution curves — that mismatch is exactly why the approximation drifts. Smaller steps, closer to the truth — but more of them Drag the slider to change the step size h . Watch the polygonal path tighten onto the true curve as h shrinks — and watch the step count climb at the same time. That's the fundamental trade-off of every numerical method: accuracy costs computation. As , the jagged path becomes indistinguishable from the true curve — but the number of steps needed grows without bound. Build the path yourself, one step at a time

This is the written version of the interactive lesson above. See the full Differential Equations course.